Tuesday, October 04, 2011

Not close enough for a mathematician

kw: mathematics, golden ratio

If you click on this image, from Wikimedia Commons, you'll get one more than twice the size, where it is easier to discern the two curves. The illustration shows the similarity between the Fibonacci Spiral (in green) and the Golden Spiral (in red). They are very close, but they are not the same.

The Fibonacci Spiral is drawn by making a set of nested squares with sides that follow the classic Fibonacci Sequence: 1, 1, 2, 3, 5, 8… In each of these nested squares you inscribe a quarter circle.

Each number in the F.S. is the sum of the prior two numbers. As you use larger and larger members of the F.S., the ratio of successive numbers approximates the Golden Ratio φ (phi) more and more closely. The value of φ is approximately 1.61803398875, and is defined as φ=(1+√5)/2. It also has the characteristic that 1/φ = φ-1.

The Golden Spiral is generated by calculating a logarithmic spiral, which is just a logarithmic curve in polar coordinates; this particular logarithmic spiral has the characteristic that the ratio of its r coordinate increases by a factor of φ with each quarter turn. This yields a factor of φ4 (6.854…) for each full turn.

At the scale shown, the two curves are difficult to distinguish. If you saw either one by itself, you'd be hard pressed to determine which one it is. The easiest way would be with a template having many circular arcs of different radii. Draw a set of axes that cross at the start of the curve, and see if a circular arc of some size matches one of the curve's sections. For the Fibonacci Spiral there is an abrupt change of curvature with each quarter turn, while the Golden Spiral has a smoothly decreasing curvature as the spiral opens up.

Logarithmic spirals and similar spirals with increasing gape as the spiral turns are rather common in nature, although a spiral that comes close to a Golden or Fibonacci spiral is quite rare. Most are tighter than this. Spirals in general have a kind of primal beauty.